Question: In Perfectville, the streets are all $20$ feet wide and the blocks they enclose are all squares of side length $400$ feet, as shown.  Sarah runs around the block on the $400$-foot side of the street, while Sam runs on the opposite side of the street.  How many more feet than Sarah does Sam run for every lap around the block?

[asy]
draw((0,0)--(10,0));
draw((1,1)--(9,1));
draw((0,0)--(0,10));
draw((0,10)--(10,10));
draw((10,10)--(10,0));
draw((9,1)--(9,9));
draw((9,9)--(1,9));
draw((1,9)--(1,1));
fill((1,1)--(1,9)--(9,9)--(9,1)--cycle,gray);
draw((9,10.5)--(9,11));
draw((10,10.5)--(10,11));
draw((9,10.75)--(10,10.75));
label("20'",(9.5,11),N);
draw((10.5,9)--(11,9));
draw((10.5,10)--(11,10));
draw((10.75,9)--(10.75,10));
label("20'",(11,9.5),E);
label("400'",(5,9),S);
label("400'",(9,5),W);
[/asy]
At each corner of the street, Sam runs 40 more feet than Sarah. Otherwise, Sam runs the same distance as Sarah. Since there are four corners, Sam runs $40\cdot4=\boxed{160}$ more feet than Sarah does.